Zermelo-Markov-Dubins problem and extensions in marine navigation

International audienceThis note accounts for optimal control techniques applied to marine navigation for seismic acquisition. More precisely, the goal is to gain time in turns and alignment maneuvers. A model for the kinematics of the marine vessel and sea current is proposed, then extended to include the evolution of the shape of the towed underwater cables during the maneuver. Two minimum time problems are stated, depending on whether the shape of the streamers is in the model or not. The simpler case is the so-called Zermelo-Markov-Dubins problem, recently studied in the literature. It generalizes the classical Dubins problem. The complete model is not standard, and preliminary analysis of controllability and of properties of minimum time trajectories are given

Zermelo-Markov-Dubins problem and extensions in marine navigation

International audienceThis note accounts for optimal control techniques applied to marine navigation for seismic acquisition. More precisely, the goal is to gain time in turns and alignment maneuvers. A model for the kinematics of the marine vessel and sea current is proposed, then extended to include the evolution of the shape of the towed underwater cables during the maneuver. Two minimum time problems are stated, depending on whether the shape of the streamers is in the model or not. The simpler case is the so-called Zermelo-Markov-Dubins problem, recently studied in the literature. It generalizes the classical Dubins problem. The complete model is not standard, and preliminary analysis of controllability and of properties of minimum time trajectories are given

Contrôle Optimal Inverse : étude théorique

This PhD thesis is part of a larger project, whose aim is to address the mathematical foundations of the inverse problem in optimal control in order to reach a general methodology usable in neurophysiology. The two key questions are : (a) the uniqueness of a cost for a given optimal synthesis (injectivity) ; (b) the reconstruction of the cost from the synthesis. For general classes of costs, the problem seems very difficult even with a trivial dynamics. Therefore, the injectivity question was treated for special classes of problems, namely, the problems with quadratic cost and a dynamics, which is either non-holonomic (sub-Riemannian geometry) or control-affine. Based on the obtained results, we propose a reconstruction algorithm for the linear-quadratic problem.Cette thèse s'insère dans un projet plus vaste, dont le but est de s'attaquer aux fondements mathématiques du problème inverse en contrôle optimal afin de dégager une méthodologie générale utilisable en neurophysiologie. Les deux questions essentielles sont : (a) l'unicité d'un coût pour une synthèse optimale donnée (injectivité); (b) la reconstruction du coût à partir de la synthèse. Pour des classes de coût générales, le problème apparaît très difficile même avec une dynamique triviale. On a donc attaqué l'injectivité pour des classes de problèmes spéciales : avec un coût quadratique, la dynamique étant soit non-holonome, soit affine en le contrôle. Les résultats obtenus ont permis de traiter la reconstruction pour le problème linéaire-quadratique

Inverse Optimal Control Problem: The Linear-Quadratic Case

International audienceA common assumption in physiology about human motion is that the realized movements are done in an optimal way. The problem of recovering of the optimality principle leads to the inverse optimal control problem. Formally, in the inverse optimal control problem we should find a cost function such that under the known dynamical constraint the observed trajectories are minimizing for such cost. In this paper we analyze the inverse problem in the case of finite horizon linear-quadratic problem. In particular, we treat the injectivity question, i.e. whether the cost corresponding to the given data is unique, and we propose a cost reconstruction algorithm. In our approach we define the canonical class on which the inverse problem is either unique or admit a special structure, which can be used in cost reconstruction

On Weyl's type theorems and genericity of projective rigidity in sub-Riemannian Geometry

18 pagesInternational audienceH. Weyl in 1921 demonstrated that for a connected manifold of dimension greater than $1$, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper, we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl projective rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl projectively rigid and genericity of Weyl projectively rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e.when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper arXiv:1801.04257[math.DG]

Turnpike Features in Optimal Selection of Species Represented by Quota Models: Extended Proofs

The paper focuses on a generic optimal control problem (OCP) deriving from the competition between two microbial populations in continuous cultures. The competition for nutrients is reduced to a two-dimensional dynamical nonlinear-system that can be derived from classical quota models. We investigate an OCP that achieves species separation over a fixed time-window, suitable for a large class of empirical growth functions commonly used in quota models. Using Pontryagin’s Maximum Principle (PMP), the optimal control strategy steering the model trajectories is fully characterized. Then, we provide sufficient conditions for the existence of a turnpike property associated with the optimal control and state-trajectories, as well as their respective co-state trajectories. Indeed, we prove that for a sufficiently large time, the optimal strategy achieving strain separation remains most of the time exponentially close to an optimal steady-state defined from an associated simpler static-OCP. This turnpike feature is based on the hyperbolicity of the linearized Hamiltonian-system around the solution of the static-OCP. The obtained theoretical results are then illustrated on microalgae, described by the Droop model in dimension 5. The optimal strategy is numerically computed in Bocop (open source toolbox for optimal control) with direct optimization methods

Turnpike features in optimal selection of species represented by quota models

International audienceThe paper focuses on a generic optimal control problem (OCP) deriving from the competition between two microbial populations in continuous cultures. The competition for nutrients is reduced to a two-dimensional dynamical nonlinear-system that can be derived from classical quota models. We investigate an OCP that achieves species separation over a fixed time-window, suitable for a large class of empirical growth functions commonly used in quota models. Using Pontryagin's Maximum Principle (PMP), the optimal control strategy steering the model trajectories is fully characterized. Then, we provide sufficient conditions for the existence of a turnpike property associated with the optimal control and state-trajectories, as well as their respective co-state trajectories. Indeed, we prove that for a sufficiently large time, the optimal strategy achieving strain separation remains most of the time exponentially close to an optimal steady-state defined from an associated simpler static-OCP. This turnpike feature is based on the hyperbolicity of the linearized Hamiltonian-system around the solution of the static-OCP. The obtained theoretical results are then illustrated on microalgae, described by the Droop model in dimension 5. The optimal strategy is numerically computed in Bocop (open source toolbox for optimal control) with direct optimization methods

Turnpike Features in Optimal Selection of Species Represented by Quota Models: Extended Proofs

The paper focuses on a generic optimal control problem (OCP) deriving from the competition between two microbial populations in continuous cultures. The competition for nutrients is reduced to a two-dimensional dynamical nonlinear-system that can be derived from classical quota models. We investigate an OCP that achieves species separation over a fixed time-window, suitable for a large class of empirical growth functions commonly used in quota models. Using Pontryagin’s Maximum Principle (PMP), the optimal control strategy steering the model trajectories is fully characterized. Then, we provide sufficient conditions for the existence of a turnpike property associated with the optimal control and state-trajectories, as well as their respective co-state trajectories. Indeed, we prove that for a sufficiently large time, the optimal strategy achieving strain separation remains most of the time exponentially close to an optimal steady-state defined from an associated simpler static-OCP. This turnpike feature is based on the hyperbolicity of the linearized Hamiltonian-system around the solution of the static-OCP. The obtained theoretical results are then illustrated on microalgae, described by the Droop model in dimension 5. The optimal strategy is numerically computed in Bocop (open source toolbox for optimal control) with direct optimization methods

Turnpike property in optimal microbial metabolite production

We consider the problem of maximization of metabolite production in bacterial cells formulated as a dynamical optimal control problem (DOCP). By Pontryagin's maximum principle, optimal solutions are concatenations of singular and bang arcs and exhibit the chattering or Fuller phenomenon, which is problematic for applications. To avoid chattering, we introduce a reduced model which is still biologically relevant and retains the important structural features of the original problem. Using a combination of analytical and numerical methods, we show that the singular arc is dominant in the studied DOCPs and exhibits the turnpike property. This property is further used in order to design simple and realistic sub-optimal control strategies

The turnpike property in maximization of microbial metabolite production

International audienceWe consider the problem of maximization of metabolite production in bacterial cells. Numerical methods showed that the major phase of the solutions for different initial states and final times is the singular regime which exhibits a special structure reminiscent of the turnpike phenomenon. We prove that singular trajectories indeed have the turnpike property by providing an estimate both on singular trajectories and on the associated controls. This result can be further used for construction of simple realistic suboptimal control strategies